Advanced Engineering Mathematics - 5th edition

Summary: Through four editions, Peter O'Neil has made rigorous engineering mathematics topics accessible to thousands of students by emphasizing visuals, numerous examples, and interesting mathematical models. ADVANCED ENGINEERING MATHEMATICS featuries a greater number of examples and problems and is fine-tuned throughout to improve the clear flow of ideas. The computer plays a more prominent role than ever in generating computer graphics used to display concepts. And problem

sets incorporate the use of such leading software packages as MAPLE. Computational assistance, exercises and projects have been included to encourage students to make use of these computational tools. The content is organized into eight-parts and covers a wide spectrum of topics including Ordinary Differential Equations, Vectors and Linear Algebra, Systems of Differential Equations, Vector Analysis, Fourier Analysis, Orthogonal Expansions, and Wavelets, Special Functions, Partial Differential Equations, Complex Analysis, and Historical Notes.

Benefits:

NEW! Throughout the text, students are invited to experiment with computations involving mathematical models and topics under discussion: for example, the effects of forcing terms and physical parameters on solution of wave and heat equations, and of filters on signal output.

NEW! Expanded treatment of special functions and orthogonal polynomials, with special attention to eigenfunction expansions and completeness of eigenfunctions leads to a discussion of the Haar wavelets.

NEW! New material on partial differential equations, including nonhomogenous heat and wave equations, the method of characteristics, and a chapter on Dirichlet and Neumann problems are included.

NEW! Expanded treatment of systems of linear and nonlinear ordinary differential equations and analysis of critical points and stability are included.

NEW! Expanded problem sets, including the introduction of sequences of problems that make use of such software packages such as MAPLE. Some problems also call on students to use computer analysis to explore relationships between solutions of mathematical models and the phenomena they model - just as they will in their careers.

NEW! More example problems are included throughout, particularly emphasizing computations based on the underlying theory.

Mathematical models of many phenomena are pursued in depth, with the objective of analyzing factors that influence behavior as well as predicting future behavior.

The computer plays a prominent role throughout the text. Computer graphics are used to display such concepts as direction field, phase portraits, surfaces and vector fields, convergence of Fourier series, the Gibbs phenomenon, and filtering noise from signals.

Summary: Through four editions, Peter O'Neil has made rigorous engineering mathematics topics accessible to thousands of students by emphasizing visuals, numerous examples, and interesting mathematical models. ADVANCED ENGINEERING MATHEMATICS featuries a greater number of examples and problems and is fine-tuned throughout to improve the clear flow of ideas. The computer plays a more prominent role than ever in generating computer graphics used to display concepts. And problem sets incorporate the use of such leading software packages as MAPLE. Computational assistance, exercises and projects have been included to encourage students to make use of these computational tools. The content is organized into eight-parts and covers a wide spectrum of topics including Ordinary Differential Equations, Vectors and Linear Algebra, Systems of Differential Equations, Vector Analysis, Fourier Analysis, Orthogonal Expansions, and Wavelets, Special Functions, Partial Differential Equations, Complex Analysis, and Historical Notes.

Benefits:

NEW! Throughout the text, students are invited to experiment with computations involving mathematical models and topics under discussion: for example, the effects of forcing terms and physical parameters on solution of wave and heat equations, and of filters on signal output.

NEW! Expanded treatment of special functions and orthogonal polynomials, with special attention to eigenfunction expansions and completeness of eigenfunctions leads to a discussion of the Haar wavelets.

NEW! New material on partial differential equations, including nonhomogenous heat and wave equations, the method of characteristics, and a chapter on Dirichlet and Neumann problems are included.

NEW! Expanded treatment of systems of linear and nonlinear ordinary differential equations and analysis of critical points and stability are included.

NEW! Expanded problem sets, including the introduction of sequences of problems that make use of such software packages such as MAPLE. Some problems also call on students to use computer analysis to explore relationships between solutions of mathematical models and the phenomena they model - just as they will in their careers.

NEW! More example problems are included throughout, particularly emphasizing computations based on the underlying theory.

Mathematical models of many phenomena are pursued in depth, with the objective of analyzing factors that influence behavior as well as predicting future behavior.

The computer plays a prominent role throughout the text. Computer graphics are used to display such concepts as direction field, phase portraits, surfaces and vector fields, convergence of Fourier series, the Gibbs phenomenon, and filtering noise from signals.

Part IV: VECTOR ANALYSIS. 11. Vector Differential Calculus. Vector Functions of One Variable. Velocity, Acceleration, Curvature and Torsion. Tangential and Normal Components of Acceleration. Curvature As a Function of t. The Frenet Formulas. Vector Fields and Streamlines. The Gradient Field and Directional Derivatives. Level Surfaces, Tangent Planes and Normal Lines. Divergence and Curl. A Physical Interpretation of Divergence. A Physical Interpretation of Curl. 12. Vector Integral Calculus. Line Integrals. Line Integral With Respect to Arc Length. Green's Theorem. An Extension of Green's Theorem. Independence of Path and Potential Theory In the Plane. A More Critical Look at Theorem 12.5. Surfaces in 3- Space and Surface Integrals. Normal Vector to a Surface. The Tangent Plane to a Surface. Smooth and Piecewise Smooth Surfaces. Surface Integrals. Applications of Surface Integrals. Surface Area. Mass and Center of Mass of a Shell. Flux of a Vector Field Across a Surface. Preparation for the Integral Theorems of Gauss and Stokes. The Divergence Theorem of Gauss. Archimedes's Principle. The Heat Equation. The Divergence Theorem As A Conservation of Mass Principle. Green's Identities. The Integral Theorem of Stokes. An Interpretation of Curl. Potential Theory in 3- Space.

Part V: FOURIER ANALYSIS, ORTHOGONAL EXPANSIONS AND WAVELETS. 13. Fourier Series. Why Fourier Series? The Fourier Series of a Function. Even and Odd Functions. Convergence of Fourier Series. Convergence at the End Points. A Second Convergence Theorem. Partial Sums of Fourier Series. The Gibbs Phenomenon. Fourier Cosine and Sine Series. The Fourier Cosine Series of a Function. The Fourier Sine Series of a Function. Integration and Differentiation of Fourier Series. The Phase Angle Form of a Fourier Series. Complex Fourier Series and the Frequency Spectrum. Review of Complex Numbers. Complex Fourier Series. 14. The Fourier Integral and Fourier Transforms. The Fourier Integral. Fourier Cosine and Sine Integrals. The Complex Fourier Integral and the Fourier Transform. Additional Properties and Applications of the Fourier Transform. The Fourier Transform of a Derivative. Frequency Differentiation. The Fourier Transform of an Integral. Convolution. Filtering and the Dirac Delta Function. The Windowed Fourier Transform. The Shannon Sampling Theorem. Lowpass and Bandpass Filters. The Fourier Cosine and Sine Transforms. The Discrete Fourier Transform. Linearity and Periodicity. The Inverse N û Point DFT. DFT Approximation of Fourier Coefficients. Sampled Fourier Series. Approximation of a Fourier Transform by an N û Point DFT. Filtering. The Fast Fourier Transform. Computational Efficiency of the FFT. Use of the FFT in Analyzing Power Spectral Densities of Signals. Filtering Noise From a Signal. Analysis of the Tides in Morro Bay. 15. Special Functions, Orthogonal Expansions and Wavelets. Legendre Polynomials. A Generating Function for the Legendre Polynomials. A Recurrence Relation for the Legendre Polynomials. Orthogonality of the Legendre Polynomials. Fourier-Legendre Series. Computation of Fourier-Legendre Coefficients. Zeros of the Legendre Polynomials. Derivative and Integral Formulas for Pn(x). Bessel Functions. The Gamma Function. Bessel Functions of the First Kind and Solutions of Bessel's Equation. Bessel Functions of the Second Kind. Modified Bessel Functions. Some Applications of Bessel Functions. A Generating Function for Jn(x). An Integral Formula for Jn(x). A Recurrence Relation for Jv(x). Zeros of Jv(x). Fourier-Bessel Expansions. Fourier-Bessel Coefficients. Sturm-Liouville Theory and Eigenfunction Expansions. The Sturm-Liouville Problem. The Sturm-Liouville Theorem. Eigenfunction Expansions. Approximation In the Mean and Bessel's Inequality. Convergence in the Mean and Parseval's Theorem. Completeness of the Eigenfunctions. Orthogonal Polynomials. Chebyshev Polynomials. Laguerre Polynomials. Hermite Polynomials. Wavelets. The Idea Behind Wavelets. The Haar Wavelets. A Wavelet Expansion. Multiresolution Analysis With Haar Wavelets. General Construction of Wavelets and Multiresolution Analysis. Shannon Wavelets.

Part VI: PARTIAL DIFFERENTIAL EQUATIONS. 16. The Wave Equation. The Wave Equation and Initial and Boundary Condition. Fourier Series Solutions of the Wave Equation. Vibrating String with Given Initial Velocity and Zero Initial Displacement. Vibrating String With Initial Displacement and Velocity. Verification of Solutions. Transformation of Boundary Value Problems Involving the Wave Equation. Effects of Initial Condition and Constants on the Motion. Wave Motion Along Infinite and Semi-Infinite String. Fourier Transform Solution of Problems on Unbounded Domains. Characteristics and d'Alembert's Solution. A Nonhomogeneous Wave Equation. Forward and Backward Waves. Normal Modes of Vibration of a Circular Elastic Membrane. Vibrations of a Circular Elastic Membrane, Revisited. Vibrations of a Rectangular Membrane. 17. The Heat Equation. The Heat Equation and Initial and Boundary Conditions. Fourier Series Solutions of the Heat Equation. Ends of the Bar Kept at Temperature Zero. Temperature in a Bar With Insulated Ends. Temperature Distribution in a Bar With Radiating End. Transformations of Boundary Value Problems Involving the Heat Equation. A Nonhomogeneous Heat Equation. Effects of Boundary Conditions and Constants on Heat Conduction. Heat Conduction in Infinite Media. Heat Conduction in an Infinite Bar. Heat Conduction in a Semi-Infinite Bar. Integral Transform Methods for the Heat Equation in an Infinite Medium. Heat Conduction in an Infinite Cylinder. Heat Conduction in a Rectangular Plate. 18. The Potential Equation. Harmonic Functions and the Dirichlet Problem. Dirichlet Problem for a Rectangle. Dirichlet Problem for a Disk. Poisson's Integral Formula for the Disk. Dirichlet Problems in Unbounded Regions. Dirichlet Problem for the Upper Half Plane. Dirichlet Problem for the Right Quarter Plane. An Electrostatic Potential Problem. A Dirichlet Problem for a Cube. The Steady-State Heat Equation for a Solid Sphere. The Neumann Problem. A Neumann Problem for a Rectangle. A Neumann Problem for a Disk. A Neumann Problem for the Upper Half Plane. 19. Canonical Forms, Existence and Uniqueness of Solutions, and Well-Posed Problems. Canonical Forms. Existence and Uniqueness of Solutions. Well-Posed Problems.

Part VII: COMPLEX ANALYSIS. 20. Geometry and Arithmetic of Complex Numbers. Complex Numbers. The Complex Plane. Magnitude and Conjugate. Complex Division. Inequalities. Argument and Polar Form of a Complex Number. Ordering. Binomial Expansion of (z = w) n. Loci and Sets of Points in the Complex Plane. Distance. Circles and Disks. The Equation |z-a| = |z-b|. Other Loci. Interior Points, Boundary Points, and Open and Closed Sets. Limit Points. Complex Sequences. Subsequences. Compactness and the Bolzano-Weierstrass Theorem. 21. Complex Functions. Limits, Continuity and Derivatives. Limits. Continuity. The Derivative of a Complex Function. The Cauchy-Riemann Equations. Power Series. Series of Complex Numbers. Power Series. The Exponential and Trigonometric Functions. The Complex Logarithm. Powers. Integer Powers. z1/n for Positive Integer n. Rational Powers. Powers zw. 22. Complex Integration. Curves in the Plane. The Integral of a Complex Function. The Complex Integral in Terms of Real Integrals. Properties of Complex Integrals. Integrals of Series of Functions. Cauchy's Theorem. Proof of Cauchy's Theorem for a Special Case. Proof of Cauchy's Theorem for a Rectangle. Consequences of Cauchy's Theorem. Independence of Path. The Deformation Theorem. Cauchy's Integral Formula. Cauchy's Integral Formula for Higher Derivatives. Bounds on Derivatives and Liouville's Theorem. An Extended Deformation Theorem. 23. Series Representations of Functions. Power Series Representations. Isolated Zeros and the Identity Theorem. The Maximum Modulus Theorem. The Laurent Expansion. 24. Singularities and the Residue Theorem. Singularities. The Residue Theorem. Some Applications of the Residue Theorem. The Argument Principle. RouchTs Theorem. Summation of Real Series. An Inversion Formula for the Laplace Transform. Evaluation of Real Integrals. 25. Conformal Mappings.

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